On the regularity of small symbolic powers of edge ideals of graphs

TL;DR

This paper investigates the regularity of symbolic powers of edge ideals of graphs, establishing bounds and exact values under various graph conditions, advancing understanding of algebraic properties linked to graph structure.

Contribution

It provides new bounds and exact formulas for the regularity of symbolic powers of edge ideals, especially relating to graph cycles and chordality, which were previously less understood.

Findings

For all s ≥ 1, reg(I(G)^{(s+1)}) is bounded by max of reg(I(G))+2s and reg(I(G)^{(s+1)}+I(G)^s).

If G has no small odd cycles, reg(I(G)^{(s)}) ≤ 2s + reg(I(G)) - 2 for s ≤ k+1.

When the complement of G is chordal, reg(I(G)^{(s)})=2s for s=2,3,4.

Abstract

Assume that $G$ is a graph with edge ideal $I(G)$ and let $I(G)^{(s)}$ denote the $s$-th symbolic power of $I(G)$. It is proved that for every integer $s\geq 1$, $${\rm reg}(I(G)^{(s+1)})\leq \max\bigg\{{\rm reg}(I(G))+2s, {\rm reg}\big(I(G)^{(s+1)}+I(G)^s\big)\bigg\}.$$As a consequence, we conclude that ${\rm reg}(I(G)^{(2)})\leq {\rm reg}(I(G))+2$, and ${\rm reg}(I(G)^{(3)})\leq {\rm reg}(I(G))+4$. Moreover, it is shown that if for some integer $k\geq 1$, the graph $G$ has no odd cycle of length at most $2k-1$, then ${\rm reg}(I(G)^{(s)})\leq 2s+{\rm reg}(I(G))-2$, for every integer $s\leq k+1$. Finally, it is proven that ${\rm reg}(I(G)^{(s)})=2s$, for $s\in \{2, 3, 4\}$, provided that the complementary graph $\overline{G}$ is chordal.